CHAPTER 4: DISCRETE RANDOM VARIABLES

What is a random variable?

A variable representing a numerical outcome of a random experiment.

What is a discrete random variable?

• A variable

• with countable possible values

• usually whole numbers

What is a probability distribution?

• A list of all possible X values

• and their probabilities

What two rules must a probability distribution follow?

Each probability is 0 to 1, and all probabilities add to 1.

What does P(X = x) mean?

The probability that X is exactly x.

What does "at most x" mean?

P(X ≤ x); include x and every value below it.

What does "at least x" mean?

P(X ≥ x);

• The probability that a random variable (X) is greater than or equal to a specific value (x)

• It includes the value (x) and every value above it.

• Ex: you take a 5 question MC quiz; what is the probability you get at least 3 correct = 3, 4, or 5

What does "fewer than x" mean?

• P(X < x)

• Strictly less than X(<x)

• do not include x

• Ex: Fewer than 5 successes means 0, 1, 2, 3, or 4 successes.

What does "more than x" mean?

P(X > x); do not include x.

What is the complement rule?

P(A) = 1 − P(not A).

What is a factorial?

Multiplying an integer by every positive integer below it.

What does 0! equal?

1.

What is the combination formula?

C(n,x) = n! / [x!(n−x)!].

When are combinations used?

When selecting items and order does not matter.

What does expected value represent?

The long-run average outcome of a random experiment.

What is the expected value formula?

E(X) = Σ[xP(X = x)].

Does expected value have to be a possible outcome?

No. It represents a long-run average.

What does variance measure?

How spread out values are around the mean.

What is the variance formula using E(X²)?

σ² = E(X²) − [E(X)]²

What is the standard deviation formula?

σ = √Variance.

What is the key idea behind binomial probability?

Count successes in a fixed number of independent trials.

What are the four binomial conditions?

1. Fixed trials

2. two outcomes

3. independence

4. constant success probability

What does X represent in binomial?

The number of successes in n trials.

What does p represent in binomial?

The probability of success on each trial.

What does q represent in binomial?

The probability of failure: q = 1 − p.

What is the binomial probability formula?

P(X = x) = [C(n,x)] × (pˣ) × (qⁿ⁻ˣ)

What is the binomial mean formula?

μ = np.

What is the binomial variance formula?

σ² = np(1−p).

What is the binomial standard deviation formula?

σ = √[np(1−p)].

How do you calculate "at least k" for binomial?

P(X ≥ k) = 1 − P(X ≤ k−1).

What is the key idea behind geometric probability?

Repeat independent trials until the first success occurs.

What conditions must a geometric experiment have?

• Two outcomes

• independence

• constant p

• continue until first success

What does X represent in the usual geometric distribution?

The number of trials needed for the first success.

What is the geometric probability formula?

P(X=x) = [(1−p)^(x−1)]p

Why is the geometric exponent x−1?

The first x−1 trials must fail before success on trial x.

What is the geometric mean formula?

μ = 1/p.

What is the geometric standard deviation formula?

σ = √(1−p) / p.

What is the key idea behind hypergeometric probability?

Count successes when sampling without replacement.

What conditions must a hypergeometric experiment have?

• Finite population

• fixed sample

• two groups

• no replacement

What does N represent in hypergeometric?

The total population size.

What does K represent in hypergeometric?

The number of successes in the population.

What does n represent in hypergeometric?

The sample size.

What does X represent in hypergeometric?

The number of successes selected in the sample.

What is the hypergeometric probability formula?

P(X=x) = [C(K,x)C(N−K,n−x)] / C(N,n).

What is the hypergeometric mean formula?

μ = n(K/N).

What is the hypergeometric variance formula?

σ² = n(K/N)(1−K/N)[(N−n)/(N−1)].

What is the key idea behind Poisson probability?

• Count events

• in a specified interval

• using an average rate

What does λ represent in Poisson?

The average number of events in the specified interval.

What does X represent in Poisson?

The number of events occurring in the specified interval.

What is the Poisson probability formula?

P(X=x) = [(e^−λ)(λ^x)] / x!.

What is the Poisson mean?

μ = λ.

What is the Poisson variance?

σ² = λ.

What is the Poisson standard deviation?

σ = √λ.

What must you do when a Poisson interval changes?

Adjust λ to match the new interval.

How do you adjust λ for a new interval?

New λ = Original λ × (New interval / Original interval).

What is the Poisson approximation formula for binomial?

λ = np.

When is Poisson used to approximate binomial?

When n is large and p is small.

What clue tells you to use binomial?

A fixed number of independent trials with constant p.

What clue tells you to use geometric?

Trials continue until the first success.

What clue tells you to use hypergeometric?

Sampling from a finite population without replacement.

What clue tells you to use Poisson?

Counting events in an interval using an average rate.